How to animate the start position object on the wave modifier?
I tried reproducing what you have described and it seems to work fine.
Upon opening the file, press Alt+A to test the animation and try moving the non animated empty to see.



We can investigate more what is causing this but you shouldn't need to apply the modifier. In fact, applying it will remove the interaction between the empty and the mesh and just turn the sphere to the shape it is under the influence permanently.

Providing more details about how did you animate the empty would also help.